Battery Lighting:
This is very easy to sort out - battery lights run from their own batteries, and you switch them on or off with a switch on the light. A few clever lights (like the B&M IX-Red Senso) have built-in light sensors so they turn themselves on automatically when it gets dark.
The advantages of battery lights are they are very simple to fit - there’s no wiring to sort out - and they’re usually removable in seconds.
The downside is you need to keep changing the batteries! Lights which use a conventional bulb will use up a set of batteries in 4 hours or so. Most rear lights now use LEDs which are much more efficient (100+ hours from the battery), and front LED lights are now appearing. Dynamo Lighting:
Dynamo systems are fiddlier to fit, but have many advantages. The biggest one is that they are always available - there are no batteries which can run down, so you always have light when you need it. A dynamo system comprises of several parts:
The Dynamo: This is the gadget which produces the power. The most common type is a bottle dynamo (like the B&M Dymotec) which has a roller running on the tyre. The other type is a hub dynamo (like the SON) which is built into the front hub. Hub dynamos are more efficient and more weatherproof, but also more work to fit and more expensive.
The Front Light: In it’s most simple form, this is just a bulb in a casing, with two terminals for the power. More advanced front lights have a Standlight system (like the B&M Lumotec+) which powers the light for a while when you stop - a brilliant safety feature. For hub dynamos, the front light has to have a switch as well, as the hub dynamo is always on (bottle dynamos are lifted off the tyre by a catch to turn them off). Front lights attach to the top of the fork, above the wheel.
The Rear Light: This is optional - you might want to use a battery LED at the back to reduce the wiring. Rear lights generally use an LED, and good ones (like the B&M DToplight+) also have the Standlight feature like the front light. Rear lights can be carrier-fitting or mudguard-fitting - though there are brackets to attach both types elsewhere if needed. Dynamo Wiring:
Dynamo lights need to be wired together - this is the trickiest bit about fitting them. Like all electrical circuits, dynamo systems have two connections - positive (+ve) and negative (-ve), also called live and earth. The trick is to get +ve and -ve from the dynamo to each light without letting them short-circuit together. There are two ways of doing this:
Frame Earth: This is the traditional way of doing it - the bicycle frame itself acts as the -ve wire, so you only have one wire to connect up. Most lights and dynamos have the -ve already connected to their mounting screws, so if the screws are tightened they make contact with the metal of the frame. The downside is that sometimes the connection doesn’t go all the way, as rear carriers and brackets sometimes don’t have a perfect electrical connection.
Two Wire: This has two wires running from the dynamo to the lights - one for +ve and one for -ve. You have two wires to connect, but they are enclosed in the same cable so you only have one wire to thread through. The advantage is that you know for certain that there is a good electrical connection for -ve. I usually use this method.
Note for Standlights and LEDs: Most lights do not care which way around they are connected - you can get +ve and -ve reversed and they will still work. But Standlights and LED lights do care, so if you find yours doesn’t work, try reversing the connections. | 
